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BULLETIN  NO.  18 


BUREAU  OF  EDUCATIONAL  RESEARCH 
COLLEGE  OF  EDUCATION 

TEACHERS'  DIFFICULTIES  IN  ARITHMETIC 
AND  THEIR  CORRECTIVES 

By 

Ruth  Streitz 

Associate,  Bureau  of  Educational  Research 


THE  LIBRARY  CF  THE 

MAR  2  3  *24 
Price   30  cents 


PUBLISHED  BY  THE  UNIVERSITY  OF  ILLINOIS,  URBANA 

1924 


-  xo 

TABLE  OF  CONTENTS 

Preface 5 

Chapter  I.    Introduction 7 

Need  for  specific  correctives 7 

Teacher  difficulties  versus  pupil  difficulties 7 

Need  that  difficulties  be  specific 7 

Purpose,  plan,  and  limitations  of  present  investigation 8 

Chapter  II.   Difficulties  Relating  to  General  Phases  of 

Arithmetic 10 

1.  Creation  of  desire  in  children  to  learn  number  facts 10 

2.  Correct  writing  of  numbers 12 

3.  Accuracy  in  copying  numbers 13 

4.  Meaning  of  figures  and  other  arithmetical  symbols 13 

5.  Recognition  of  numbers 14 

Chapter  III.   Difficulties  Encountered  in  Operations  of 

Arithmetic 15 

6.  Fluency  in  addition 15 

7.  Fluency  in  the  fundamental  operations 16 

8.  Concept  of  subtraction 17 

9.  Borrowing  process  in  subtraction 18 

10.  Proficiency  in  subtraction  in  the  intermediate  and 

upper  grades 19 

11.  Multiplication  tables,  customary  sequence 19 

12.  Multiplication  tables,  miscellaneous  combinations 22 

13.  Relationship  between  multiplication  and  division 23 

14.  Division  of  uneven  numbers  by  2 24 

15.  Confusion  in  division  forms 24 


57&25S 


16.  Accuracy  in  determination  of  quotient  figures  in  long 
division 24 

17.  Reduction  of  common  fractions 25 

18.  Multiplication  of  common  fractions 26 

19.  Division  of  common  fractions 26 

20.  Accuracy  in  placing  decimal  point 27 

Chapter  IV.    Difficulties  Relating  to  Denominate  Num- 
bers and  Problem  Solving 29 

21.  Presentation  of  denominate  numbers 29 

22.  Square  measure 29 

23.  Cubic    measure 30 

24.  Lack  of  self-reliance  of  pupil  in  solving  problems 30 

25.  Solving  of  problems 31 

26.  Accurate  statements  of  problems 32 

27.  Reading  of  problems 33 

28.  Oral  explanation  of  problems 33 


PREFACE 

Observation  of  teachers  at  work  and  conferences  with  them 
indicate  that  some  are  highly  effective  in  handling  certain 
phases  of  their  work  which  others  find  difficult.  Although 
all  teachers  may  not  be  equally  efficient  in  the  application  of 
a  given  method  and  teaching  device,  the  pooling  of  successful 
practice  should  provide  a  store  of  procedures  to  which  teach- 
ers could  turn  for  assistance  in  handling  difficulties  which 
they  encounter.  This  bulletin  reports  the  results  of  an 
attempt  to  collect  successful  teaching  procedures  in  the  field 
of  arithmetic.  The  list  is  doubtless  far  from  complete  but  the 
bulletin  is  published  with  the  hope  that  teachers  in  arithmetic 
will  find  it  helpful. 

The  technique  employed  in  carrying  on  this  study  has 
not  been  used  extensively  and  the  experience  of  Miss  Streitz 
should  be  of  interest  to  one  contemplating  an  investigation  of 
this  type.  One  obstacle  encountered  is  the  fact  that  few 
teachers  have  analyzed  their  teaching  so  that  they  know 
what  their  difficulties  are.  This  condition  is  significant  both 
from  the  standpoint  of  educational  research  and  with  respect 
to  efficient  teaching. 

This  report  represents  the  cooperation  of  a  large  number  of 
teachers.  To  all  who  have  contributed,  the  Bureau  of  Educa- 
tional Research  gratefully  acknowledges  its  indebtedness. 

Walter  S.  Monroe,  Director. 
February  13,  1924. 


Digitized  by  the  Internet  Archive 

in  2011  with  funding  from 

University  of  Illinois  Urbana-Champaign 


http://www.archive.org/details/teachersdifficul18stre 


TEACHERS'  DIFFICULTIES  IN  ARITHMETIC 
AND  THEIR  CORRECTIVES 

CHAPTER  I 
INTRODUCTION 

Need  for  specific  correctives.  A  vast  amount  of  literature  on 
the  curriculum  and  general  principles  of  teaching  is  now  available. 
Our  attention  has  been  called  to  the  large  individual  differences 
which  exist  in  most  classes  and  much  emphasis  has  been  given  to 
adapting  instruction  to  the  various  levels  of  intelligence.  General 
principles  of  teaching  are,  however,  inadequate  for  attaining  the 
highest  degree  of  efficiency.  Even  when  teachers  are  skillful  in 
applying  general  methods  they  not  infrequently  obtain  unsatisfactory 
results  with  some  or  all  members  of  a  certain  class.  Hence,  they  feel 
a  need  for  correctives  to  deal  with  specific  difficulties.  Within  recent 
years  considerable  attention  has  been  given  to  the  identification  of 
specific  difficulties  and  the  correctives  required.1 

Teacher  difficulties  versus  pupil  difficulties.  The  teacher's  task 
is  to  stimulate  and  direct  the  pupil  in  his  learning.  In  assisting  the 
pupil  to  overcome  a  particular  difficulty  which,  as  a  learner,  he  may 
have  encountered,  the  teacher  herself  may  or  may  not  meet  with 
difficulties  also.  Her  difficulty,  however,  although  related  to  the 
pupil's,  is  not  identical  with  it.  In  addition,  the  teacher  because  of 
the  general  conduct  of  her  class  may  have  to  contend  with  obstacles 
which  have  no  counterpart  in  the  experiences  of  the  pupils. 

Need  that  difficulties  be  specific.  The  teacher  may  be  aware  that 
results  are  unsatisfactory.  For  example,  Johnny  can  not  work  his  long 
division  examples  readily,  or  is  unable  to  solve  written  problems  and 
Henry  does  not  like  arithmetic.  When  expressed  in  this  way  the 
difficulty  is  general  rather  than  specific,  and  in  order  to  be  dealt  with 
effectively  it  must  be  defined.  Johnny's  general  difficulty  with  long 
division  may  be  due  to  a  lack  of  proficiency  in  subtraction  or  to  the 
determination  of  the  figures  in  the  quotient  or  to  particular  cases 

*Gray,  William  Scott.  "Remedial  cases  in  reading:  their  diagnosis  and  treat- 
ment." Supplementary  Educational  Monograph  No.  22.  Chicago:  University  of 
Chicago,  1922. 

Terry,  Paul  Washington.  ''How  numerals  are  read."  Supplementary  Educa- 
tional Monograph  No.  18.   Chicago:  University  of  Chicago,  1922. 

[7] 


such  as  the  placing  of  a  zero  in  the  quotient.  Edith  may  be  unable  to 
solve  problems  because  she  does  not  know  the  meaning  of  certain 
technical  words  used  in  the  statement,  or  she  may  not  be  sufficiently 
acquainted  with  the  practical  situation  from  which  the  problem  is 
taken.  Henry's  dislike  for  arithmetic  may  be  due  to  any  one  or  a 
combination  of  a  number  of  causes.  Such  causes  must  be  analyzed 
into  specific  elements  before  they  can  be  dealt  with  efficiently.  This 
constitutes  the  teacher's  difficulty  and  should  be  considered  an  im- 
portant phase  of  her  work.  "Every  teacher  should  resort  to  the 
method  of  analysis  as  it  is  only  by  this  means  that  the  teacher  is 
able  to  apply  proper  instruction  so  that  the  pupil  may  be  helped 
to  improve."2 

In  many  instances  teachers  have  not  determined  the  cause  of  the 
trouble  and  yet  pupils  have  been  able  to  advance  in  spite  of  the 
errors  they  have  made.  Any  degree  of  success  attained  in  such  cases 
is  due  probably  to  ways  devised  by  the  pupils  themselves  for  meeting 
their  needs  and  not  to  conscious  help  on  the  part  of  the  teacher. 
Such  instances  are  far  more  numerous  than  we  have  been  led  to 
think.  One  of  the  most  striking  examples  of  such  a  procedure  was 
reported  by  Uhl.3  The  use  of  standardized  tests  showed  a  boy  weak 
in  subtraction  but  unusually  strong  in  multiplication.  By  inquiring 
into  the  method  which  he  used  it  was  found  that  when  asked  to 
subtract  he  turned  the  example  into  one  of  multiplication.  If  he  had 
to  subtract  9  from  46,  he  took  46,  set  aside  1  so  as  to  secure  a 
number  that  would  be  an  exact  multiple  of  9.  He  then  disintegrated 
his  45  into  five  9's  and  dropped  one  of  the  9's,  thus  performing  the 
required  subtraction.  The  reason  for  the  boy's  failure  is  obvious. 
The  teacher,  so  long  as  the  correct  answer  could  be  given,  had  not 
taken  pains  to  find  out  exactly  how  the  work  was  accomplished. 

Purposes  of  investigation.  The  first  purpose  of  the  investigation 
reported  in  this  bulletin  was  to  compile  a  list  of  specific  difficulties 
which  teachers  are  actually  encountering  in  the  field  of  arithmetic. 
The  second  purpose  was  to  formulate  for  each  difficulty  one  or  more 
proven  correctives.  These,  for  the  most  part,  have  been  restricted  to 
correctives  which  teachers  are  using  successfully. 

Plan  of  investigation.  During  the  school  year  of  1921-22  a  re- 
quest was  addressed  to  the  superintendents  of  city  school  systems  in 

2Judd,  Charles  H.  "Analysis  of  learning  processes  and  specific  teaching," 
Elementary  School  Journal,  21:655-64,  May,  1921. 

3Uhl,  W.  L.  "The  use  of  standardized  materials  in  arithmetic^  for  diagnosing 
pupils'  methods  of  work,"  Elementary  School  Journal,   18:215-18,  November,   1917. 

[8] 


Illinois,  asking  them  to  invite  their  teachers  to  report  specific  diffi- 
culties which  they  were  encountering  in  the  teaching  of  arithmetic.  The 
responses  to  this  invitation  were  used  to  compile  a  tentative  list  of 
specific  difficulties.  The  second  step  in  the  investigation  was  a  visit 
to  a  number  of  school  systems  for  the  purpose  of  observing  teaching 
in  arithmetic,  and  of  interviewing  teachers  in  regard  to  the  specific 
difficulties  which  they  were  experiencing.  Correctives  for  various 
specific  difficulties  were  formulated  also  during  this  visitation.  In 
most  instances  the  investigator  asked  the  superintendent  to  direct  her 
to  some  of  his  most  successful  teachers  of  arithmetic.  Often  these 
teachers  were  kind  enough  to  demonstrate  their  teaching  of  a  par- 
ticular topic  which  was  not  a  part  of  the  regular  work.  In  order  to 
become  familiar  with  the  general  principles  governing  the  teaching 
of  the  subject  and  to  secure  additional  devices  for  correcting  specific 
difficulties,  books  and  articles  dealing  with  the  teaching  of  arithmetic 
were  also  consulted. 

Limitations  of  the  method  of  investigation.  The  outstanding 
limitation  of  the  investigation  was  the  inability  of  teachers  to  define 
their  difficulties.  Many  teachers  even  asserted  that  they  experienced 
no  difficulties  in  the  teaching  of  arithmetic;  others  were  able  to  men- 
tion only  general  items,  some  of  which  related  to  classroom  manage- 
ment rather  than  to  actual  teaching;  still  others  reported  such 
matters  as  "how  to  secure  interest"  and  "how  to  hold  attention."  A 
number  of  primary  teachers  stated  difficulties  relating  to  a  particular 
textbook,  complaining  that  the  children,  because  of  the  complexity  of 
the  subject  matter,  were  not  able  to  advance  beyond  a  certain  point. 
Altogether,  the  total  number  of  specific  difficulties  collected  is  much 
smaller  than  anticipated.  In  the  writer's  opinion,  this  is  due  largely 
to  the  fact  that  teachers  were  not  able  to  analyze  and  define  their 
difficulties. 

The  determination  of  the  correctives  is  empirical  rather  than 
scientific.  No  attempt  has  been  made  to  bring  together  a  complete 
list  of  correctives  for  each  difficulty.  Neither  have  the  correctives 
listed  been  tested  experimentally,  but  they  are  believed  to  be  in 
agreement  with  generally  accepted  educational  principles. 

Plan  of  report.  In  the  following  chapters  specific  difficulties  are 
treated  separately.  A  definite  statement  of  the  difficulty  itself  is  first 
given,  followed  usually  by  a  brief  explanation.  In  succeeding 
paragraphs,  under  the  head  of  "correctives"  one  or  more  explicit 
methods  of  dealing  with  this  difficulty  are  described. 

[9] 


CHAPTER  II 

DIFFICULTIES  RELATING  TO  GENERAL  PHASES 

OF  ARITHMETIC 

Difficulty  1.   How  to  create  in  young  children  a  desire  to  learn  the 
number  facts  which  they  need. 

Although  this  difficulty  was  mentioned  by  a  number  of  teachers 
the  exact  meaning  is  not  clear.  The  number  facts  which  a  child 
actually  needs  in  his  own  activities  are  very  different  from  those 
which  the  teacher  thinks  he  will  need  at  some  future  time.  The 
meaning  of  the  word  "young"  is  also  indefinite  but  in  the  following 
discussion  will  be  interpreted  as  referring  to  children  in  the  primary 
grades. 

Correctives:  1.  Delaying  teaching  of  number  work.  It  is  cus- 
tomary in  many  schools  to  begin  formal  instruction  in  arithmetic  in 
the  first  grade.  This  is  due  in  part  to  tradition  but  doubtless  also  to 
the  fact  that  it  is  relatively  easy  to  set  exercises  in  arithmetic.  As  a 
result,  children  are  frequently  asked  to  learn  many  number  facts 
before  they  have  encountered  any  need  for  them  in  their  activities. 
The  simplest  corrective  would  be  to  delay  such  teaching  altogether 
until  children  experience  the  need  for  it.  This  suggestion  is  in  agree- 
ment with  the  opinions  held  by  many  prominent  educators. 

2.  Need  for  number  work  created  in  school  situations.  A  pri- 
mary teacher  may  very  properly  manipulate  the  school  life  of  the 
child  so  that  he  will  feel  a  need  for  number  facts  in  his  daily  activ- 
ities. Extremely  artificial  situations  should  be  avoided  as  far  as 
possible,  but  there  are  many  devices  which  seem  natural  and  which 
will  probably  prove  interesting  and  enjoyable  to  the  pupils. 

a.  Games  as  a  means  of  creating  need.  Children  like  to  play 
games  which  call  for  some  knowledge  of  number  facts.  For  example, 
counting  and  matching  contests  may  be  arranged  with  dominoes  and 
loto;  simple  work  in  addition  and  multiplication  is  needed  in  bean- 
bag  and  ring-toss.  Any  game  which  requires  that  scores  be  kept 
creates  a  desire  for  some  phase  of  number  work. 

b.  Reproduction  of  adult  activities  as  creating  need.  Many 
teachers  find  the  reproduction  of  certain  adult  activities  helpful  in 
creating  a  need  for  various  phases  of  number  work.    The  use  of 

[10] 


these  devices  is  not  confined  to  the  primary  grades.  For  example, 
playing  store  is  of  great  interest  to  children.  It  is  just  as  much  fun 
to  be  the  purchaser  as  it  is  to  be  the  storekeeper  and  both  involve  a 
certain  amount  of  understanding  of  number  facts.  The  making  of 
change,  adding  of  pennies,  using  pint  and  quart  measures,  and  the 
counting  of  objects  are  all  a  part  of  the  play.  Some  teachers  have 
a  "store"  arranged  in  one  corner  of  the  room  which  serves  as  a  basis 
for  many  types  of  activities.  The  "store"  generally  consists  of  several 
shelves,  and  a  counter  over  which  supplies  are  bought  and  sold. 
Upon  the  shelves  may  be  found  articles  in  their  original  wrappings, 
such  as  baking  powder  cans,  breakfast  foods  of  different  kinds,  coffee, 
tea,  soap,  etc.  The  appearance  of  these  articles  so  arranged  is  not 
unlike  a  corner  of  a  real  grocery  store.  Perishable  articles  as  fruit 
and  vegetables  are  sometimes  moulded  in  clay  and  painted  with  water 
colors.  The  money  used  should  resemble  real  money,  if  possible,  and 
when  given  in  exchange  for  the  article  purchased,  the  pupil  should 
know  the  denomination  of  the  coin  and  its  purchasing  power.  An 
understanding  of  the  meaning  of  numbers  in  such  a  situation  takes 
place  quite  naturally  and  the  child's  interest  is  much  greater  than  it 
could  possibly  be  if  the  material  were  presented  in  a  remote  or 
artificial  environment.  The  teacher  should  not  lose  sight  of  the  fact 
that  a  device  of  this  kind  is  only  a  means  to  an  end,  and  should  be 
used  only  when  there  is  need  for  securing  a  stronger  motive  or  for 
interesting  pupils  in  certain  phases  of  number  work. 

c.  School  activities  as  a  means  of  creating  need.  Some  teachers 
have  found  it  helpful  to  utilize  certain  school  activities  as  a  means  of 
causing  children  to  feel  a  need  for  arithmetic.  For  example,  when 
health  campaigns  or  health  crusades  were  started  in  various  school 
systems,  the  serving  of  milk  followed  in  order  that  children  might  be 
helped  in  attaining  the  correct  weight  for  their  height.  Milk  was 
delivered  each  morning  at  the  different  schools  and  each  child  who 
could  pay  a  certain  amount  was  served.  This  necessitated  bringing 
money  from  home  to  purchase  the  week's  supply.  In  Urbana,  one 
teacher  looked  upon  the  serving  of  milk  as  an  opportunity  for  in- 
cidental teaching  of  number  work.  The  children  know  how  many 
pennies  to  bring  in  order  to  secure  milk  for  the  entire  week.  If  a 
holiday  occurs  they  know  that  no  milk  will  be  served  and  the  price 
of  one  day's  milk  will  be  deducted..  Children  like  to  handle  money 
and  like  the  feeling  of  responsibility  which  accompanies  the  making 
of  purchases. 

[11] 


A  banking  system  maintained  in  the  schools  may  be  an  out- 
growth of  a  thrift  campaign.  Quincy  has  a  "banking  day"  when  the 
children  bring  their  pennies  to  be  deposited  in  one  of  the  banks.  As 
each  child  comes  up  to  the  teacher's  desk  with  his  bank  book  and 
pennies  the  teacher  asks  how  much  money  he  has,  how  many  pennies 
in  a  nickle,  how  many  nickles  in  a  dime,  etc.  In  this  way  the  child 
gains  an  idea  of  certain  number  facts.  Although  these  serve  to 
acquaint  children  with  number  facts  the  amount  of  time  required  to 
credit  each  child's  account  is  questioned.  Is  there  enough  real  educa- 
tional material  presented  by  this  method  to  justify  such  an  expendi- 
ture of  time?  Much  would  depend  upon  the  system  of  banking  as 
well  as  the  plan  which  the  teacher  follows. 

Difficulty  2.   How  to  teach  children  to  write  numbers  correctly. 

This  difficulty  is  found  most  frequently  in  the  primary  grades 
although  in  some  cases  children  in  the  intermediate  and  even  in  the 
grammar  grades  have  not  learned  how  to  write  certain  numbers 
correctly.  It  is  not  uncommon  for  some  children  to  write  figures 
backwards;  figure  3  is  sometimes  written  £,  6  is  written  d,  and  9  is 
written  p.  There  is  also  a  tendency  to  reverse  figures  and  when 
attempting  to  write  12,  children  will  write  21,  etc.  Such  errors, 
however,  are  not  common  to  all  children;  with  some  there  is  little 
confusion  regarding  the  form  or  arrangement  of  figures. 

Corrective.  Errors  in  writing  numbers  are  due  to  the  absence  of 
certain  specific  habits.  In  most  cases  there  has  been  a  lack  of  suffi- 
cient practice.  All  learning  is  a  matter  of  growth  and  children  must 
have  time  in  which  to  learn  even  those  things  which  seem  very 
simple.  There  are  several  devices  which  may  be  resorted  to  in  help- 
ing the  child  overcome  this  difficulty.  One  teacher  of  the  lower 
grades  had  several  children  who  seemed  unable  to  write  3  correctly. 
She  placed  a  row  of  3's  on  the  board  and  beside  each  3  was  a 
"straight  line"  or  tooth-pick  drawing  of  a  man  facing  in  the  same 
direction  as  the  3's.  The  teacher  suggested  that  the  3's  and  the  little 
men  were  soldiers  who  always  faced  in  the  right  direction.  The 
children  compared  their  figures  with  the  illustration  and  could  easily 
recognize  the  correct  and  the  incorrect  forms. 

Another  teacher  called  attention  to  the  difference  between  6  and  9 
and  suggested  that  in  writing  the  9  they  think  always  of  the  way  the 
balloon  man  holds  his  balloons  when  offering  them  for  sale.  The  top 
of  the  9  is  like  a  balloon  on  the  end  of  a  string.  Later  on  the  children 

[12] 


are  able  to  make  6  correctly  by  remembering  to  invert  the  balloon. 
Nearly  everyone  develops  an  individual  system  for  remembering 
things  which  seems  to  aid  in  making  correct  responses  to  certain 
situations.  Any  device  which  is  not  too  remote  in  its  resemblance  to 
the  difficulty  in  question  is  justifiable  if  it  facilitates  in  any  way  the 
overcoming  of  that  difficulty. 

Difficulty  3.   How  to  secure  accuracy  in  copying  numbers. 

Children  are  required  to  copy  numbers  in  working  the  examples 
and  problems  given  in  the  textbook.  They  also  have  to  copy  numbers 
in  making  calculations.  The  fact  that  children  are  surprisingly  in- 
accurate in  copying  is  probably  due  to  the  relatively  little  attention 
which  is  given  to  this  phase  of  arithmetic  work.  Many  teachers  ap- 
pear to  assume  that  the  process  is  so  simple  that  it  does  not  need 
to  be  taught. 

Corrective.  Teachers  should  give  explicit  training  in  copying 
figures.  It  is  customary  for  them  to  give  attention  to  the  writing  of 
figures  from  dictation,  but  this  will  not  suffice.  There  should  be 
definite  exercises  in  which  pupils  are  asked  to  copy  figures  from  the 
blackboard  or  textbook.  It  is  advisable  for  teachers  to  prepare 
simple  tests  in  which  both  the  rate  and  accuracy  of  copying  will  be 
measured. 

Difficulty  4.  How  to  teach  the  meaning  of  figures  and  other  symbols. 
Some  children  have  trouble  in  associating  appropriate  meanings 
with  numbers  and  other  symbols  which  are  used  in  arithmetic.  This 
difficulty  is  confined  for  the  most  part  to  the  primary  grades  but 
occasionally  it  persists  even  after  the  pupil  has  passed  beyond  the 
third  grade. 

Corrective.  The  attention  of  the  child  should  be  directed  to  the 
"number  families"  of  20,  30,  40,  etc.  A  large  calendar  also  serves  to 
acquaint  the  children  with  numbers  in  a  very  useful  way.  This  might 
be  referred  to  daily  until  the  children  have  learned  to  recognize  the 
figures  wherever  they  may  be  found.  A  "digit  tree"  drawn  on  the 
board  or  made  in  the  form  of  a  poster  is  a  delight  to  young  children; 
the  numbers  up  to  10  are  placed  on  the  various  branches  and  because 
zero  is  nothing  it  has  no  place  in  the  branches  but  must  remain  on 
the  ground.  The  teacher  should  write  in  both  the  figures  and  the 
words,  the  two  being  used  together  so  that  the  children  will  see  their 
relationship.    For  example,  "22"  and  "twenty-two"  should  appear 

[13] 


together  until  the  child  has  learned  that  one  is  just  another  way  of 
writing  the  other.  There  is  no  need,  however,  to  write  out  figures  of 
large  denominations,  for  before  using  numbers  of  such  magnitude, 
as  350,623  or  even  1,725,  the  child  will  have  advanced  sufficiently  in 
his  school  work  to  understand  their  meaning  in  words. 

Difficulty  5.   How  to  teach  children  to  "see  numbers"  correctly. 

This  relates  to  the  difficulty  which  some  children  experience  in 
the  perception  of  a  figure.  They  are  unable  to  pick  out  distinguishing 
characteristics  so  that  they  are  able  to  recognize  the  figures  easily. 

Corrective.  Children  generally  develop  a  system  of  their  own  to 
aid  in  the  recognition  of  the  number  but  the  teacher  can  be  of  assist- 
ance in  calling  attention  to  its  formation.  She  could  point  to  the  turn 
in  the  figure  "8"  and  mention  that  it  is  like  the  beginning  of  the 
letter  "s."  In  one  room  visited  the  children  were  writing  4's  while 
the  teacher  kept  saying  down,  over,  down,  with  a  certain  marked 
rhythm.  This  no  doubt  directed  the  attention  of  the  children  to  the 
formation  of  the  number  and  aided  in  a  later  recognition. 


[14] 


CHAPTER  III 

DIFFICULTIES  ENCOUNTERED  IN  THE  OPERATIONS 
OF  ARITHMETIC 

Difficulty  6.   How  to  teach  addition. 

This  difficulty,  as  well  as  the  corresponding  ones  for  subtraction, 
multiplication,  division,  and  fractions,  is  not  specific.  Study  of  arith- 
metical abilities  has  shown  that  there  are  a  number  of  specific  abil- 
ities in  each  of  these  divisions  of  arithmetic.  Thus  the  general 
ability  implied  in  the  question  "How  to  teach  addition"  should  be 
broken  up  into  specific  difficulties,  which  will  in  general  probably 
correspond  to  the  various  types  of  addition  examples.1 

Corrective.  The  first  step  then  in  securing  fluency  in  addition  is 
to  break  up  the  general  difficulty  into  specific  ones  corresponding  to 
the  various  types  of  addition  examples.  The  second  step  is  to  make 
definite  and  adequate  provision  for  teaching  each  type  of  example. 
Drill  in  addition  must  be  appropriately  distributed  among  the  several 
types.  As  an  illustration,  the  most  effective  procedure  for  securing 
proficiency  in  column  addition  is  by  drill  upon  column  addition 
itself  and  not  upon  the  combinations.  Similarly  in  teaching  addition 
which  involves  carrying,  drill  upon  such  examples  must  be  given. 
Even  in  the  teaching  of  the  combinations  it  is  not  sufficient  to  drill 

4  7 

pupils  only  upon  7  ,  but  also  upon  4  .   In  general,  the  combinations 

IT  IT 

involving  the  larger  digits  are  more  difficult  and  require  more  prac- 
tice. Usually  this  will  mean  that  the  teacher  will  need  to  exert  special 
effort  because  it  has  been  shown  that  in  general,  textbooks  present  a 
much  larger  number  of  exercises  on  the  simpler  combinations.  The 
simpler  combinations  also  are  taught  earlier  when  there  is  a  longer 
period  of  practice. 

There  are  many  effective  devices  for  giving  drill  upon  number 
combinations  and  other  types  of  examples.  Some  teachers  keep  a 
list  of  simple  examples  upon  the  blackboard  and  devote  a  few 
minutes  each  day  to  work  upon  them.  In  Jacksonville,  a  fourth-grade 

'For  a  list  of  the  types  of  examples  in  addition  see 

Monroe,   Walter  S.    Measuring  the   Results  of  Teaching.    Boston:     Houghton 
Mifflin  Company,  1918,  p.  111-12. 

[15] 


teacher  was  observed  to  spend  about  two  minutes  each  day  with 
Thompson's  Minimum  Essential  Practice  Material.  For  this  purpose, 
Studebaker  Arithmetic  Exercises  and  the  Courtis  Standard  Practice 
Tests  are  also  suitable.  In  practice  upon  the  combinations,  after 
the  first  stages  of  learning,  there  is  no  fixed  order.  Each  combina- 
tion should  be  mastered  so  thoroughly  that  the  sight  of  it  calls  forth 
the  correct  answer.  The  arrangement  of  the  addition  facts  in  table 
form  is  merely  a  device  which  is  useful  in  the  beginning. 

In  the  lower  grades  teachers  were  observed  to  make  much  use 
of  concrete  examples  in  teaching  the  fundamental  combinations  and 
in  giving  practice  upon  other  types  of  examples.  In  connection  with 
this  corrective,  it  should  be  pointed  out  that  a  problem  is  not  made 
concrete  by  giving  numbers  such  labels  as  apples,  dollars,  pounds, 
etc.;  it  must  be  identified  by  the  child  with  his  own  experience.  Thus 
problems  which  are  given  a  concrete  interpretation  in  this  way  are 
frequently  useful  for  teaching  the  concept  of  an  operation.  They 
also  furnish  practice  upon  the  operation  but  there  will  be  need 
probably  always  for  some  formal  drill  exercises.  One  teacher  reported 
that  concrete  problems  were  inefficient  for  drill  purposes  because  the 
children  had  difficulty  in  deciding  just  which  operations  were 
called  for. 

Difficulty  7.  How  to  secure  fluency  in  the  fundamental  operations. 

This  difficulty  might  be  thought  of  as  being  included  in  the 
preceding  one  but  a  number  of  teachers  mentioned  it  separately.  In 
expressing  this  difficulty  accuracy  was  usually  given  more  emphasis 
than  rapid  work  but  both  concepts  were  expressed. 

Corrective.  In  securing  fluency  there  should  be  emphasis  upon 
rapid  work.  Practice  should  be  timed.  This  can  be  done  most 
effectively  when  the  pupils  are  provided  with  printed  or  mimeo- 
graphed lists  of  exercises.  Scores  in  terms  of  both  the  number  of 
examples  attempted  and  the  number  of  examples  correct  should  be 
kept.  If  it  is  desired  to  give  special  emphasis  to  accuracy  the  penalty 
for  errors  may  be  increased. 

In  case  mimeographed  or  printed  practice  exercises  are  not  avail- 
able appropriate  examples  may  be  placed  on  the  blackboard.  If 
space  permits,  the  list  may  be  duplicated  so  that  two  or  more  pupils 
can  be  working  at  the  same  time.  In  one  school  two  pupils  were 
sent  to  the  board.  They  stood  with  their  backs  to  the  examples  until 
the  signal  to  begin  was  given.   They  then  began  work  and  the  one 

[16] 


who  finished  first  was  declared  the  winner,  and  was  allowed  to  com- 
pete with  another  pupil.  The  contest  feature  of  this  drill  creates  a 
keen  interest.  In  the  case  of  addition  the  answers  may  be  erased 
and  the  same  exercises  used  by  other  pupils.  The  rules  can  be 
adjusted  to  give  appropriate  emphasis  to  rate  of  work  and  to 
accuracy. 

Difficulty  8.   How  to  teach  the  concept  of  subtraction. 

A  distinction  is  made  here  between  the  concept  of  subtraction 
and  skill  in  working  subtraction  examples.  Although  an  under- 
standing of  the  meaning  of  subtraction  is  usually  a  prerequisite  for 
skill  in  the  operation,  it  will  not  insure  the  possession  of  such  skill. 

Corrective.  There  are  two  methods  which  have  been  widely 
used  in  explaining  the  process  of  subtraction:  (1)  the  additive 
method,  and  (2)  the  take-away  method.  Pupils  are  familiar  with 
addition  when  they  reach  the  topic  of  subtraction.  It  would  there- 
fore seem  logical  to  connect  the  two  processes  as  closely  as  possible. 
Our  observations  and  the  testimony  of  several  teachers  have  been 
to  the  effect  that  when  the  additive  method  is  used  in  the  lower 
grades  the  pupils  discard  it  for  the  take-away  method  in  the  upper 
grades.  A  number  of  teachers  expressed  the  conviction  that  because 
of  this  the  take-away  method  should  be  used  from  the  beginning. 
Beatty2  states  that  pupils  using  the  additive  method  are  more 
accurate  in  subtraction  but  work  more  slowly.  He  concludes  that 
although  the  additive  method  seems  to  be  the  more  logical  procedure 
its  use  is  open  to  question.  He  states  also  that  the  exclusive  use  of 
either  method  is  not  justified. 

It  seems  likely  that  the  additive  method  is  useful  in  explaining 
the  process  of  subtraction  to  children  who  are  having  difficulty  with 
the  take-away  method  but  our  observations  in  this  investigation 
justify  the  conclusion  that  subtraction  should  be  taught  by  the 
take-away  method.  In  any  case,  one  method  should  be  adopted  by 
the  school  and  followed  by  all  teachers  who  give  instruction  in  sub- 
traction. "Nothing  insures  confusion  more  certainly  than  the  indis- 
criminate use  of  different  methods  in  different  grades."3 

In  teaching  subtraction  by  the  take-away  method  the  use  of 
objects  which  may  be  counted  is  helpful.    Blocks,  splints,  pencils, 


2Beatty,  Willard  W.  "The  additive  versus  the  borrowing  method  in  subtraction," 
Elementary  School  Journal,  21:198-200,  November,  1920. 

sLennes,  N.  J.  The  Teaching  of  Arithmetic.  New  York:  Macmillan  Company, 
1923,  p.  227. 

[17] 


books,  etc.  may  be  used.  If  the  child  is  given  five  objects  and  told 
to  take  away  two,  he  can  count  the  remainder. 

Difficulty  9.   How  to  teach  the  borrowing  process  in  subtraction. 

The  process  of  borrowing  in  subtraction  is  one  of  the  first  serious 
difficulties  which  pupils  encounter  in  learning  arithmetic.  The 
present  tendency  to  teach  addition  and  subtraction  together  so  as  to 
make  clear  to  the  pupil  that  the  process  of  subtraction  is  the  reverse 
of  addition  will  probably  tend  to  lessen  the  seriousness  of  this 
difficulty. 

Corrective.  One  teacher  suggested  that  this  difficulty  was  due  to 
confusion  concerning  the  value  of  zero.  One  of  her  pupils  stated  that 
zero  was  more  than  nine,  and  in  support  of  the  assertion  turned  to 
his  arithmetic  and  pointed  out  that  the  digits  were  printed  in  the 
following  order:  1,  2,  3,  4,  5,  6,  7,  8,  9,  0.  Since  0  was  given  after  9 
the  natural  inference  was  made  that  it  was  greater  than  9.  This  con- 
fusion could  easily  be  avoided  by  having  "0"  precede  1  whenever  the 
digits  are  presented  in  their  natural  sequence. 

In  teaching  the  borrowing  process,  the  small  figures  resulting 
from   borrowing   may   be   written   above   the   minuend   as    follows: 

7  13  18 

846  This  device,  however,  is  merely  a  "crutch"  which  should  be 
289 

used  only  in  the  earlier  stages  of  learning.  It  is  obvious  that  a  pupil 
can  not  work  subtraction  examples  rapidly  by  this  procedure.  In  some 
instances  where  teachers  have  permitted  the  continued  use  of  this 
"crutch,"  the  individual  has  been  unable,  even  in  adult  life,  to  sub- 
tract quickly  and  accurately. 

A  teacher  in  East  St.  Louis  reported  an  ingenious  device  for  ex- 
plaining borrowing  which  she  had  found  useful.  In  illustrating  this 
device  she  used  the  example  6000 — 3876.  Units,  tens,  hundreds  and 
so  forth  were  spoken  of  as  families.  Each  one  borrowed  only  from 
the  family  immediately  to  the  left.  The  particular  article  borrowed 
was  biscuits.  In  this  example  units  had  no  biscuits  for  dinner;  they 
tried  to  borrow  from  tens  and  tens  then  had  to  borrow  from 
hundreds  and  hundreds  from  thousands.  The  peculiar  thing  about 
this  borrowing  was  that  thousands  would  not  lend  less  than  one  pan 
of  biscuits.  When  hundreds  received  this  pan  and  attempted  to  put 
it  into  their  pans  they  had  enough  for  ten  pans.  They  loaned  one 
of  these  pans  to  tens,  leaving  them  nine,  and  the  process  was  repeated 
for  the  other  places.    As  a  result  there  were  finally  ten  biscuits  in 

[18] 


units  place,  nine  pans  in  tens,  nine  pans  in  hundreds,  and  five  pans 
in  thousands.  This  explanation  was  presented  as  an  attractive  nar- 
rative and  pupils  were  required  to  answer  questions,  relative  to  the 
number  of  remaining  biscuits,  at  various  stages  of  the  explanation. 

Because  of  failure  to  understand  the  process  of  borrowing,  some 
children  do  not  realize  that  they  can  not  take  a  larger  number  from 
a  smaller.  For  example,  such  pupils  will  not  hesitate  to  subtract  846 
from  673.  The  fact  that  they  are  able  to  borrow  in  units  place  and 
perform  the  subtraction  doubtless  is  responsible  for  their  attempt  to 
do  likewise  in  hundreds  place.  This  particular  difficulty  can  be  over- 
come by  the  use  of  concrete  problems.  Most  pupils  will  readily 
understand  that  they  can  not  take  $800  from  $600  when  the  example 
is  expressed  in  this  form. 

Difficulty  10.    How  to  secure  proficiency  in  subtraction  in  the  in- 
termediate and  upper  grades. 

Although  corresponding  difficulties  exist  for  the  other  three 
fundamental  operations  it  appears  that  the  case  of  subtraction  is 
peculiar.  In  some  of  the  schools  visited,  pupils  in  the  intermediate 
and  grammar  grades  were  observed  to  do  satisfactory  work  in  all 
the  fundamental  operations  except  subtraction.  In  some  cases  they 
were  especially  defective  in  this  one  operation. 

Corrective.  In  correcting  this  difficulty  it  should  be  recognized 
that  instruction  in  the  operations  of  arithmetic  can  not  be  completed 
in  the  primary  grades.  In  fact,  it  can  not  be  entirely  completed  by 
the  end  of  the  intermediate  grades.  The  teacher  in  each  grade 
should  recognize  that  she  has  a  distinct  responsibility,  as  it  is  only 
human  for  the  pupils  to  forget  and  to  lose  skill.  Even  when  the 
teaching  in  the  lower  grades  has  been  excellent,  additional  training 
will  probably  be  required  each  successive  year  in  order  to  maintain 
a  high  degree  of  fluency.  It  is  highly  important  that  the  pupils  be 
given  the  right  start  and  that  the  instruction  in  the  lower  grades  be 
efficient,  but  this  difficulty  can  not  be  corrected  entirely  by  these 
means. 

Difficulty  11.   How  to  teach  the  multiplication  tables. 

This  is  one  of  the  traditional  difficulties  in  the  field  of  arithmetic. 
Some  of  the  harder  tables  have  proven  a  serious  stumbling  block  to 
many  pupils.  For  some  reason  the  multiplication  tables  constitute 
a  more  serious  difficulty  than  the  corresponding  tables  in  the  other 
operations. 

[19] 


Corrective.  The  writer  believes  that  one  of  the  reasons  why  the 
multiplication  tables  so  frequently  constitute  a  difficulty  is  that  the 
teaching  of  them  is  spread  out  over  too  long  a  period  of  time.  It 
seems  reasonable  that  when  the  learning  of  the  tables  is  spread  out 
over  several  months  or  even  over  more  than  one  year,  as  is  true  in 
some  cases,  the  pupils  fail  to  recognize  it  as  a  task  to  be  undertaken 
and  completed.  If  this  thesis  is  true,  one  corrective  would  be  to  focus 
the  attention  of  the  pupils  on  the  multiplication  tables  and  to  com- 
press the  teaching  of  them  into  two  or  three  weeks. 

Some  teachers  made  the  comment  that  pupils  were  not  interested 
in  learning  the  multiplication  tables.  This  is  probably  due  largely  to 
the  fact  that  they  are  bored  with  the  slow  pace  expected  of  them  in 
many  schools.  Children  usually  like  to  be  given  a  definite  task  and 
to  concentrate  upon  it.  They  are  less  capable  of  sustained  interest 
than  adults.  In  most  cases,  they  will  take  an  interest  in  something 
which  appeals  to  them  as  constituting  a  real  challenge  to  their 
abilities. 

Because  of  the  importance  which  teachers  appear  to  attach  to 
this  difficulty,  suggestions  are  given  below  for  the  teaching  of  the 
various  tables.  However,  it  should  be  borne  in  mind  that  after  the 
first  stages  of  learning  the  pupil  should  receive  practice  upon  the 
combinations  in  miscellaneous  order. 

1.  The  number  2  in  multiplication.  Counting  by  2's  is  useful  in 
building  up  the  table.  The  pupils  seated  in  one  row  of  seats  may  be 
asked  to  hold  up  both  hands,  then  other  pupils  walk  down  the  aisle 
counting  by  2's.  Others  may  be  arranged  two  abreast  in  a  march  and 
counted  by  2's.  Such  imaginary  games  as  the  mailing  of  letters  where 
the  cost  of  stamps  must  be  computed,  or  the  buying  of  certain 
articles  as  pencils,  plums,  etc.  at  two  cents  each,  might  prove  helpful. 
One  teacher  suggested  the  use  of  pint  and  quart  measures  in  teaching 
multiplication  by  the  number  2.  There  are  two  pints  in  one  quart  and 
pupils  may  be  asked  how  many  pints  in  two  quarts  of  milk,  three 
quarts  of  milk,  etc. 

2.  The  number  3  in  multiplication.  Counting  by  3's  is  likewise 
used  in  teaching  the  table  of  3's.  A  foot  rule  and  a  yardstick  may  be 
used  in  much  the  same  way  as  the  pint  and  quart  for  the  table  of 
2's.  Simple  concrete  problems  can  be  easily  formulated. 

3.  The  number  4  in  multiplication.  The  suggestions  already 
made  for  counting  and  for  the  use  of  concrete  examples  apply  to  the 
table  of  4's.    One  teacher  suggested  the  use  of  the  fact  that  a  horse 

[20] 


needs  four  shoes.  Pupils  may  be  asked  to  tell  how  many  shoes  are 
needed  for  two  horses,  three  horses,  etc.  The  use  of  the  denominate 
number  relation  between  quarts  and  gallons  is  also  recommended. 

4.  The  number  5  in  multiplication.  The  table  of  5's  is  perhaps 
the  easiest  and  generally  no  special  devices  are  needed  for  teaching 
it.  Concrete  problems  involving  the  purchase  of  articles  at  five  cents 
each  will  be  found  useful.  Problems  involving  street-car  fares  at  five 
cents  each  have  also  been  suggested. 

5.  The  number  6  in  multiplication.  The  table  of  6's  may  be 
illustrated  by  the  purchase  of  articles  by  the  half  dozen.  The  use  of 
six  working  days  in  a  week  was  suggested.  Problems  such  as  "How 
many  days  will  a  man  work  in  three  weeks,  five  weeks,  etc?"  may 
be  used. 

6.  The  number  7  in  multiplication.  The  table  of  7's  was  men- 
tioned frequently  as  the  one  causing  the  greatest  difficulty.  One 
teacher  made  this  illuminating  comment:  "Children  are  so  tired  of 
the  multiplication  tables  by  the  time  the  table  of  7's  is  reached  that 
it  is  very  hard  to  get  them  interested  enough  to  learn  the  facts." 
This  comment  serves  to  reinforce  the  thesis  presented  on  page  20. 
If  the  facts  in  the  multiplication  tables  were  taught  as  one  large  piece 
of  work  to  be  accomplishd  in  a  few  days  and  not  strung  out  over 
several  weeks  or  months  this  tendency  of  the  children  to  lose  interest 
would  be  avoided.  Furthermore,  it  may  be  pointed  out  that  when 
the  tables  involving  the  higher  numbers  are  delayed  the  pupils 
receive  less  drill  upon  the  combinations  because  there  is  less  time. 
The  greater  amount  of  the  practice  period  has  been  given  to  the 
easier  combinations.  If  the  idea  of  multiplication  has  been  ration- 
alized in  the  teaching  of  the  preceding  tables,  there  will  be  little  need 
for  the  use  of  concrete  problems  in  connection  with  the  teaching  of 
the  table  of  7's  and  of  the  two  following.  The  derivation  of  the  multi- 
plication combinations  in  the  case  of  higher  tables  will  be  largely  a 
waste  of  time. 

7.  The  number  8  in  multiplication.  The  comments  just  made 
with  reference  to  the  table  of  7's  also  apply  to  the  table  of  8's. 
Furthermore,  it  should  be  noted  that  after  the  tables  up  to  8  have 
been  learned,  the  table  of  8's  introduces  only  two  new  combina- 
tions, 8X8  and  8X9,  provided  the  pupils  have  been  taught  both 
forms  of  a  given  combination,  e.  g.,  7x4  =  28  and  4x7  =  28. 

[21] 


8.  The  number  9  in  multiplication.  The  table  of  9's  should  not 
constitute  a  real  difficulty.  The  pupil  should  know  all  of  the  facts 
except  9x9.  (If  the  tables  beyond  the  9's  are  to  be  taught  this 
statement  would  need  to  be  modified  accordingly.)  Practice  should 
now  be  given  involving  all  of  the  combinations  in  miscellaneous 
order.  Various  games  and  devices  can  be  used.  The  pupil  by  this 
time  has  reached  the  place  where  the  emphasis  should  be  given  to 
securing  fluency  in  the  use  of  multiplication  combinations. 

Difficulty  12.  How  to  teach  the  multiplication  tables  so  that  pupils 

will  be  skillful  in  using  combinations  in  other  than  the  cus- 
tomary sequence. 

This  difficulty  is  really  included  in  the  preceding  one  since  the 
tables  should  always  be  taught  so  that  pupils  will  be  skillful  in  using 
the  combinations  in  other  than  the  ordinary  sequence.  No  teaching 
of  this  topic  can  be  considered  satisfactory  which  does  not  engender 
this  ability  in  the  pupils. 

Corrective.  After  the  first  stages  in  the  learning  of  the  multipli- 
cation tables,  drill  upon  the  combinations  should  be  in  miscellaneous 
order  rather  than  in  the  customary  sequence.  Skill  will  come  only 
with  practice.  So  far  as  possible,  this  practice  should  be  given  by 
doing  exercises  similar  to  those  in  which  the  pupils  will  use  the 
combinations.  Drill  upon  the  combinations  alone  will  not  make 
pupils  proficient  in  doing  more  complex  types  of  multiplication 
examples. 

A  number  of  teachers  were  observed  using  effective  devices  for 
giving  drill  on  the  combinations.  Among  the  best  noted  are  the 
following.4 

1.  Number  races.  One  type  of  a  number  race  is  to  select  two 
children  to  go  to  the  board,  each  equipped  with  a  pointer.  A  list  of 
multiplication  combinations  are  written  on  the  board.  Their  class- 
mates take  turns  in  naming  products  of  these  combinations  and  the 
two  at  the  board  answer  by  pointing  to  the  combination  which 
belongs  with  the  product.  Each  child  tries  to  be  the  first  to  touch  the 
right  number  with  the  pointer.  The  game  can  be  varied  by  having 
the  products  written  on  the  board  and  the  pupils  at  their  seats  name 
the  combinations. 


*For  other  games  and  similar  devices  see, 

Smith,    David    Eugene    and    others.     "Number    games    and    number   rhymes," 
Teachers  College  Record,  8:1-3,  November,  1912. 

[22] 


A  further  variation  of  this  game  may  be  secured  by  dividing  the 
class  into  two  teams.  In  addition  one  pupil  should  be  chosen  as  a 
score  keeper.  One  pupil  from  each  team  is  sent  to  the  board  and 
each  pupil  in  his  seat  then  asks  for  one  number.  After  all  have 
participated  two  other  children  are  sent  to  the  board  and  the  process 
is  repeated.  A  score  of  one  point  is  given  for  each  time  a  child  at  the 
board  is  the  first  to  point  to  the  correct  answer,  and  the  team  receiv- 
ing the  greater  number  of  points  wins  the  game. 

2.  Bean  bags.  (For  use  in  the  primary  grades.)  Draw  a  circle 
on  the  floor  like  the  one  in  the  picture.  Choose  sides  with  a  leader 
for  each  side.  The  game  consists  in  throwing  the  bean  bags  at  the 
circle  (Fig.  1)  and  in  striking  the  largest  numbers  possible.  The 
leader  has  the  first  turn,  then  each  child  comes  forward  for  his  turn. 
The  score  equals  the  number  struck  multiplied  by  itself.  When  a 
line  is  struck  the  score  is  zero. 

3.  Multiplication  game.  This  game  is  similar  to  the  bean  bag 
game  except  that  you  multiply  the  number  in  the  circle  by  those 
around  the  rim  (Fig.  2).  One  child  can  be  selected  to  point  to  the 
numbers  while  the  children  in  the  different  rows  perform  the 
multiplication. 


/\  5 

/ 

/     6 ■   >v 

3    \ 

I    e>  / 

%      I 

\S    4 

7 

Fig.  1  Fig.  2 

Difficulty  13.   How  to  get  pupils  to  understand  the  close  relation- 
ship which  exists  between  multiplication  and  division. 

The  occurrence  of  this  difficulty  is  probably  due  to  the  fact  that 
multiplication  and  division  are  treated  as  separate  topics. 

Corrective.  As  in  other  instances,  pupils  will  be  unlikely  to  grasp 

[23] 


the  relationship  between  multiplication  and  division  unless  it  is 
definitely  taught.  Thus  to  overcome  the  difficulty  one  should  make 
specific  provision  for  teaching  this  relationship.  One  suggestion  is 
to  teach  multiplication  and  division  simultaneously.  Such  a  pro- 
cedure would  probably  serve  to  engender  an  understanding  of  the 
relationship  but  it  is  not  unlikely  that  it  would  interfere  with  the 
effective  teaching  of  the  multiplication  tables.  After  the  pupils  have 
grasped  the  idea  of  multiplication  and  have  become  familiar  with 
at  least  some  of  the  combinations  it  is  not  inappropriate  to  introduce 
the  idea  of  division,  particularly  in  dealing  with  concrete  problems. 
When  the  pupils  have  solved  a  problem  in  which  multiplication  is 
required  they  may  be  given  an  inverse  problem  calling  for  division. 

Difficulty  14.  How  to  teach  pupils  to  divide  an  uneven  number  by  2. 

This  difficulty  was  mentioned  frequently  by  primary  teachers 
but  probably  does  not  apply  to  intermediate  and  grammar  grades. 

Corrective.  One  teacher  stated  that  she  met  this  difficulty  by 
calling  the  one  left  over  a  remainder.  To  express  it  as  a  fraction 
appears  to  be  confusing  to  pupils  and  in  the  primary  grades  there  is 
little  or  nothing  to  be  gained  by  insisting  upon  the  expression  of  the 
complete  quotient. 

Difficulty  15.   How  to  avoid  confusion  due  to  the  variety  of  forms 
by  which  division  is  indicated. 

There  are  at  least  four  ways  in  which  division  is  indicated: 

56 


42  -J-  7  =     ,       8)128(,      87  |  3654,     -y     . 

Teachers  report  that  pupils  tend  to  be  confused  when  these  methods 
of  indicating  division  are  presented  together. 

Corrective.  The  obvious  corrective  is  to  use  only  those  forms  of 
expressing  division  that  are  necessary.  The  second  and  fourth  forms 
can  easily  be  avoided.  Several  teachers  mentioned  that  pupils  were 
confused  because  different  teachers  used  different  forms  in  long 
division.  This  suggests  that  it  is  highly  desirable  for  teachers  in  the 
successive  grades  to  agree  upon  the  same  form. 

Difficulty  16.  How  to  secure  accuracy  in  the  determination  of  quo- 
tient figures  in  long  division. 

Pupils  make  many  errors  in  the  determination  of  quotient  fig- 
ures. One  teacher  mentioned  that  many  of  her  pupils  set  down  a 
quotient  figure  which  was  too  small  and  as  a  result  secured  a  remain- 
der larger  than  the  divisor,  and  that  they  failed  to  recognize  this  as 

[24] 


an  indication  of  a  mistake  in  their  work.  Another  type  of  error  is 
the  omission  of  zeros  in  the  quotient.  Occasionally  pupils  make  errors 
due  to  bringing  down  two  figures  of  the  dividend  at  one  time.  Teach- 
ers generally  appear  to  agree  that  long  division  constitutes  one  of 
the  most  serious  difficulties  in  the  field  of  arithmetic. 

Corrective.  One  corrective  which  was  suggested  is  to  train  pupils 
in  estimating  quotients  and  to  insist  that  this  is  done  before  the  actual 
work  of  division  is  started.  If  pupils  are  skillful  in  estimating  quo- 
tients, gross  errors  can  be  easily  detected. 

There  is  not  just  one  difficulty  in  long  division;  there  are  several. 
Investigation  has  indicated  that  there  are  several  types  of  examples, 
each  of  which  presents  a  distinctive  difficulty.5  For  example,  an  ex- 
ercise which  results  in  such  a  quotient  as  508  involves  a  difficulty  due 
to  the  presence  of  a  zero  in  the  quotient.  Another  difficulty  is  encoun- 
tered when  there  is  a  remainder.  Still  other  difficulties  grow  out  of 
the  relation  of  the  trial  divisor  to  the  dividend.  Explicit  training 
should  be  given  on  each  type  of  example.  It  is  doubtless  true  that 
many  pupils  experience  difficulty  in  long  division  because  in  the 
training  which  they  have  received  one  or  more  types  of  examples 
have  been  neglected. 

Difficulty  17.   How  to  teach  reduction  of  common  fractions. 

The  topic  of  common  fractions  has  suffered  in  somewhat  the 
same  way  as  that  of  the  multiplication  tables.  Fractions  are  com- 
monly introduced  very  early  in  the  child's  educational  career  and 
each  year  he  encounters  examples  which  are  slightly  more  difficult. 
There  are  certain  advantages  in  this  plan  but  there  is  this  weakness 
also;  the  pupil  fails  to  approach  the  study  of  fractions  with  the  atti- 
tude that  it  constitutes  a  real  task  and  demands  the  concentration  of 
his  attention.  Some  of  the  difficulties  in  handling  fractions  could 
doubtless  be  overcome  by  singling  them  out  as  specific  tasks  and  by 
concentrating  upon  them. 

Corrective.  In  connection  with  the  reduction  of  fractions  pupils 
should  be  made  familiar  with  the  principle  that  "fractions  may  be 
changed  in  form,  without  altering  their  values,  by  multiplying  or 
dividing  both  terms  by  the  same  number."  If  the  meaning  of  this 
principle  is  made  clear  through  appropriate  illustrations  there  should 
be  frequent  reference  to  it  in  the  reduction  of  fractions. 

The  reduction  of  fractions  to  the  lowest  common  denominator 


6Monroe,  Walter  S.    Measuring  the  Results  of  Teaching.    Boston:    Houghton 
Mifflin  Company,  1918,  p.  112-13. 

[25] 


was  specifically  mentioned.  As  a  partial  corrective  for  this  difficulty 
it  may  be  pointed  out  that  it  is  not  imperative  that  the  common  de- 
nominator be  the  lowest  one.  Any  denominator  which  is  a  multiple 
of  each  of  the  denominators  will  be  satisfactory;  the  lowest  is  desira- 
ble only  as  a  means  for  reducing  the  necessary  calculations  to  a 
minimum.  In  some  cases  the  reduction  of  fractions  to  a  common 
denominator  has  been  made  artificially  difficult  by  insisting  upon  the 
lowest  common  denominator. 

Difficulty  18.   How  to  teach  multiplication  of  common  fractions. 

Most  of  the  difficulties  in  common  fractions  occur  when  one  of 
the  factors  is  an  integer  or  a  mixed  number.  The  multiplication  of 
a  fraction  by  a  fraction  does  not  usually  constitute  a  serious  difficulty 
but  the  multiplication  of  a  mixed  number  by  a  mixed  number  was 
mentioned  as  being  especially  difficult. 

Corrective.  A  fundamental  principle  which  should  be  recognized 
in  overcoming  this  difficulty  is  to  recognize  each  type  of  example  and 
to  single  it  out  for  specific  instruction.  Training  upon  the  multipli- 
cation of  one  fraction  by  another  fraction  will  help  pupils  very  little 
in  multiplying  one  mixed  number  by  another.  The  presentation  of 
more  than  one  procedure  for  doing  examples  will  probably  tend  to 
confuse  most  pupils.  Some  teachers  recommend  that  both  integers 
and  mixed  numbers  should  be  reduced  to  improper  fractions  and 
then  the  multiplication  may  be  performed  in  exactly  the  same  way 
as  when  the  product  of  two  fractions  is  required. 

Difficulty  19.   How  to  teach  division  of  common  fractions. 

Most  of  the  difficulties  encountered  in  the  division  of  common 
fractions  center  around  the  inversion  of  the  divisor.  The  relative 
magnitude  of  the  divisor  and  dividend  is  also  a  source  of  trouble. 
Many  pupils  are  confused  when  the  divisor  is  greater  than  the  divi- 
dend. 

Corrective.  One  means  of  correcting  this  difficulty  which  pupils 
encounter  is  to  rationalize  the  process  of  division,  i.  e.,  train  pupils  to 
think  through  the  whole  process  step  by  step.  Although  this  sugges- 
tion has  many  advocates,  it  is  not  inappropriate  to  raise  the  question, 
"Is  it  desirable  to  rationalize  the  topic  of  division  of  fractions?"  Suz- 
zallo  has  stated  that  "The  rationalization  of  a  process  which  should 
be  performed  mechanically  is  merely  to  stir  up  unnecessary  trouble, 
trouble  unprompted  by  a  demand  of  actual  efficiency."6   This  state- 

6Suzzallo.    Henry.    The   Teaching  of   Primary   Arithmetic.    Boston:    Houghton 
Mifflin  Co.,  1912,  p.  64-65. 

[26] 


merit  appears  to  be  particularly  applicable  to  the  topic  of  division  of 
fractions.  The  process  should  be  performed  mechanically  and  there 
is  little  or  no  advantage  in  understanding  the  reason  for  each  step. 
It  seems  likely  that  the  most  effective  way  of  controlling  this  diffi- 
culty is  by  training  the  pupils  to  invert  the  divisor  and  then  multiply 
without  attempting  any  explanations  of  the  rule. 

A  number  of  teachers  have  found  it  helpful  to  analyze  the  writ- 
ten work  of  pupils  for  the  purpose  of  determining  the  types  of  errors 
which  they  most  frequently  make.  In  such  studies  it  has  been  found 
that  certain  types  occur  much  more  often  than  others.  When  a  teacher 
knows  the  errors  which  her  class  tends  to  make  she  is  able  to  focus 
her  instruction  upon  their  correction. 

Difficulty  20.   How  to  secure  accuracy  in  placing  the  decimal  point 
in  products  and  quotients. 

It  may  be  pointed  out  that  aside  from  the  reading  and  writing 
of  decimal  fractions,  this  is  the  only  serious  difficulty  which  this 
topic  involves.  Otherwise  decimal  fractions  may  be  treated  much 
as  integers. 

Corrective.  Investigation7  has  indicated  that  pupils  tend  to  place 
decimal  points  by  means  of  specific  rules  rather  than  by  a  single 
general  rule.  If  this  is  the  case  there  should  be  explicit  training  in 
each  type  of  example.  Some  drill  of  course  is  provided  when  an 
example  involving  decimal  fractions  is  worked  but  it  is  advisable  to 
single  out  the  activity  of  placing  the  decimal  point  and  drill  upon 
that.  This  can  be  done  by  using  examples  with  products  and  quo- 
tients given  correctly  except  for  the  decimal  point.  The  pupils  may 
then  be  required  to  insert  the  decimal  point  in  the  correct  place.  The 
following  are  types  of  exercises  which  have  been  found  useful: 


657.2 
.7 

Multiplication 
Division 

67.50 
.03 

46004 

20250 

.47 

|  2758.9     Answer 

587 

8.2  [38754    Answer  47 

In  the  case  of  multiplication  the  pupil  merely  inserts  the  decimal 

'Monroe,  Walter  S.    ''The  ability  to  place  the  decimal  point  in  division,"  Ele- 
mentary School  Journal,   18:287-93,  December,   1917. 

[27] 


point.    In  the  division  exercises  he  is  required  to  write  the  quotient 
in  the  appropriate  place  and  to  insert  the  decimal  point. 

Another  corrective  which  is  used  by  some  teachers  is  to  require 
pupils  to  estimate  the  product  or  quotient.  This  procedure  will  tend 
to  prevent  gross  errors.  Checking  the  work  is  also  a  means  of  de- 
tecting errors  but  in  general  will  require  considerable  time. 


[28] 


CHAPTER  IV 

DIFFICULTIES  RELATING  TO  DENOMINATE  NUMBERS 
AND  PROBLEM  SOLVING 

Difficulty  21.  How  to  present  the  study  of  denominate  numbers  so 
that  pupils  will  be  interested  in  the  topic  and  will  remember 
the  facts. 

As  stated  this  is  a  general  difficulty.  In  practice,  it  is  likely  that 
pupils  may  be  interested  in  some  denominate  numbers  and  not  in 
others.  Thus  the  following  discussion  of  correctives  should  be  thought 
of  as  applying  to  those  denominate  numbers  which  given  groups  of 
pupils  do  not  find  interesting. 

Corrective.  In  the  earlier  grades,  interest  in  tables  of  denomi- 
nate numbers  can  be  secured  by  utilizing  practical  problems  which 
arise  in  the  regular  work  of  the  school  and  by  playing  such  games  as 
store,  bank,  etc.  The  teacher  may  appropriately  manipulate  the  ac- 
tivities of  the  children  so  that  a  need  for  denominate  numbers  will 
be  created. 

In  the  intermediate  grades  where  denominate  numbers  are 
studied  more  systematically  the  use  of  concrete  problems  will  serve 
to  engender  interest  in  many  pupils.  It  is  advisable  to  show  meas- 
uring instruments  such  as  a  foot  rule,  yardstick,  pint  measure,  gallon 
measure,  bushel  measure,  etc.  Some  teachers  have  pupils  measure  a 
liquid  in  order  to  determine  the  relation  between  pint  and  quart, 
quart  and  gallon,  etc.  If  the  pupil  is  given  a  pint  measure  and  a 
quart  measure  he  will  find  that  two  pints  of  water  will  fill  the  quart 
measure.  Information  can  be  derived  in  the  same  way  from  the  quart 
measure  and  the  gallon  measure. 

Another  means  of  overcoming  this  difficulty  is  by  eliminating 
from  the  course  of  study  all  obsolete  or  obsolescent  measures.  It 
appears  that  frequently  interest  is  lessened  because  pupils  are  asked 
to  learn  tables  of  denominate  numbers  which  are  no  longer  used. 

Difficulty  22.  How  to  make  pupils  understand  that  the  units  of 
square  measure  are  used  to  measure  areas. 

Pupils  have  difficulty  in  identifying  the  table  of  square  measure 
as  a  means  of  describing  areas.    They  are  also  confused  over  the 

[29] 


products  of  the  dimensions  of  a  rectangle  being  expressed  in  terms 
of  a  square  unit. 

Corrective.  Doubtless  one  cause  for  this  difficulty  is  the  insist- 
ence that  the  multiplier  is  always  an  abstract  number  and  that  the 
product  must  be  of  the  same  denomination  as  the  multiplicand.  Al- 
though there  is  logical  justification  for  this  principle,  for  practical 
purposes  we  may  justify  the  conflicting  principle  that  feet  multiplied 
by  feet  gives  square  feet  and  that  the  area  expressed  in  terms  of 
square  units  is  the  product  of  the  two  dimensions. 

As  in  the  case  of  many  other  difficulties,  the  fundamental  rule 
to  be  observed  is  to  single  out  the  difficulty  and  give  specific  atten- 
tion to  overcoming  it.  In  the  field  of  arithmetic  where  the  outcome 
of  instruction  consists  largely  of  habits,  repetition  must  be  provided 
for. 

Difficulty  23.   How  to  teach  cubic  measure. 

The  difficulty  in  the  case  of  cubic  measure  is  similar  to  that 
mentioned  above  for  square  measure. 

Corrective.  It  is  desirable  to  have  a  number  of  blocks  whose 
dimensions  are  one  inch.  With  a  sufficient  number  of  these  blocks  a 
cubic  foot  can  be  built  up  and  parallelepipeds  of  various  dimensions 
can  be  constructed.  Pupils,  who  have  difficulty  in  understanding  the 
relation  between  cubic  content  and  the  unit  of  measurement,  may 
then  count  the  number  of  cubic  inches  in  a  cubic  foot  or  in  the  par- 
ticular parallelepiped  under  construction.  They  will  probably  grasp 
the  idea  that  a  convenient  way  of  counting  these  blocks  is  to  ascer- 
tain the  number  in  the  top  layer  and  then  to  multiply  this  by  the 
number  of  layers.  In  this  way  the  pupil  should  easily  grasp  the  rule 
that  volume  is  equal  to  the  product  of  length  by  width  by  depth 
or  height. 

Difficulty  24.   How  to  overcome  the  pupil's  lack  of  self-reliance  in 
solving  problems. 

A  lack  of  self-reliance  is  frequently  indicated  by  the  pupils  when 
he  erases  all  of  his  work  and  begins  anew  on  the  solution.  Some 
pupils  "give  up."  Still  others  are  content  to  accept  any  answer  which 
they  obtain. 

This  difficulty  is  not  easy  to  overcome,  because  the  trait  ex- 
hibited functions  in  other  activities.  In  many  cases  it  is  character- 
istic of  the  pupil.  The  teacher  faces  the  problem  of  causing  the  pupil 
to  acquire  self-reliance   not  only  in  problem  solving  but  in   other 

[30] 


phases  of  his  school  work  and  even  in  the  things  which  he  does  out- 
side of  school. 

Corrective.  In  our  emphasis  upon  the  need  for  training  pupils 
to  study,  it  has  been  observed  that  teachers  should  not  give  too  much 
assistance.  If  pupils  do  not  have  an  opportunity  to  face  their  diffi- 
culties and  to  overcome  them  they  will  never  become  self-reliant. 
It  is  sometimes  advisable  to  allow  pupils  to  make  mistakes  because 
they  need  the  experience  of  determining  the  corrections.  They  should 
be  given  time  to  think.  The  solving  of  problems  is  something  which 
can  not  be  done  in  a  mechanical  way.  Timid  pupils  are  frequently 
discouraged  because  they  are  not  given  a  chance  to  complete  their 
problems.  The  teacher's  task  is  to  stimulate  and  direct  the  learning 
activities  of  his  pupils;  and  whenever  he  does  any  of  the  work  for 
them  he  thereby  deprives  them  of  an  opportunity  to  learn.  It  is 
likely  that  he  also  tends  to  destroy  their  self-reliance. 

Difficulty  25.   How  to  teach  pupils  to  solve  problems. 

Many  pupils  fail  to  grasp  the  idea  of  problem  solving.  Some  of 
them  think  of  it  as  a  mechanical  application  of  certain  rules  or  the 
duplication  of  a  procedure  illustrated  in  a  problem  worked  out  in  the 
text.  Still  others  think  of  it  as  a  trial  and  error  process  in  which 
they  perform  certain  operations  upon  certain  numbers  in  the  hope 
that  they  will  secure  an  answer  which  will  be  accepted  as  correct. 

Corrective.  A  fundamental  principle  which  should  be  recognized 
in  dealing  with  this  difficulty  is  that  pupils  must  be  trained  to  solve 
problems  rather  than  to  secure  correct  answers.  The  emphasis  should 
constantly  be  placed  upon  learning  how  to  solve  problems  rather 
than  upon  the  answers  obtained.  A  pupil  may  learn  a  great  deal 
about  the  solving  of  problems  even  when  he  is  failing  to  secure  a 
correct  answer.  Furthermore,  by  doing  a  very  small  number  of  ex- 
ercises, he  may  learn  much  about  the  solving  of  problems. 

In  training  pupils  to  solve  problems  the  teacher  should  have 
clearly  in  mind  the  steps  of  the  process. 

(1)  The  first  step  is  to  read  the  statement  of  the  problem.  In 
this  statement  there  are  two  kinds  of  words,  first,  those  which  de- 
scribe the  setting  of  the  problem  or  the  particular  environment  in 
which  it  occurs,  and  second,  those  which  define  quantities  or  quan- 
titative relationships.  The  second  class  of  words  may  be  called 
"technical"  and  precise  meanings  must  be  associated  with  them.  The 
reading  of  a  problem  is  a  complex  process  and  generally  a  higher 

[31] 


degree  of  comprehension  is  required  than  in  our  reading  of  ordinary 
printed  material. 

(2)  The  principles  applicable  to  the  problem  must  be  recalled. 
For  example,  in  the  problem  "A  man  invests  $893  in  property.  He 
sells  the  property  for  $1050.  What  is  his  rate  of  profit?",  it  is  neces- 
sary for  the  pupil  to  recall  the  principle  that  the  rate  of  profit  is  cal- 
culated upon  the  amount  invested,  and  not  upon  the  selling  price. 

(3)  The  meaning  of  the  technical  words  and  the  principles  re- 
called are  used  in  formulating  a  plan  of  procedure  for  solving  the 
problem.  In  doing  this  one  must  not  overlook  words  which  may 
appear  inconspicuous  in  the  statement  of  the  problem  but  which, 
nevertheless,  are  important  in  specifying  the  relationships  between 
the  quantities. 

(4)  The  plan  of  solution  must  be  verified.  In  some  cases  this 
verification  may  be  made  before  calculations  are  begun.  Sometimes 
the  pupil  will  be  unable  to  verify  his  plan  until  he  has  worked  the 
problem  through. 

The  performance  of  the  operations  specified  in  the  plan  of  solu- 
tion is  not  a  part  of  the  reasoning  process.  The  reflective  thinking 
has  been  completed  when  the  plan  of  solution  has  been  formulated. 

Difficulty  26.  How  to  train  pupils  to  make  accurate  statements. 

Many  of  the  errors  in  arithmetic  are  due  to  permitting  inaccu- 
rate statements  of  problems  in  the  classroom.  One  authority  has  said 
that,  "It  is  the  loose  manner  of  writing  out  solutions,  tolerated  by 
many  teachers  that  gives  rise  to  half  the  mistakes  in  reasoning  which 
vitiate  the  pupils'  work  and  teachers  are  coming  to  realize  that  inac- 
curacies of  statement  tend  to  beget  inaccuracy  of  thought  and  so 
should  not  be  tolerated  in  the  classroom."1 

Corrective.  It  is  likely  that  careless  reading  of  problems  and  of 
explanations  is  one  cause  for  inaccurate  statements  by  pupils.  In- 
sistence upon  correct  reading  will  do  much  to  overcome  this  difficulty. 
The  teacher  should  also  insist  that  the  pupils  attach  precise  meanings 
to  the  technical  terms  used  in  the  statements  of  problems  and  in  the 
explanation  of  the  procedures  employed  in  their  solution.  Investiga- 
tion has  shown  that  pupils  use  many  words  without  being  sufficiently 
acquainted  with  their  meaning. 

The  correction  of  the  tendency  to  make  inaccurate  statements  is 
to  be  accomplished  in  much  the  same  way  as  the  correction  of  other 

1Brown,  J.  C,  and  Coffman,  L.  D.  How  to  Teach  Arithmetic.  Chicago:  Row, 
Peterson  and  Co.,  1914,  p.  44-45. 

[32] 


errors.    There  must  be  painstaking  attention  to  details  on  the  part 
of  the  teacher  and  no  errors  should  be  permitted  to  pass  unnoticed. 

Difficulty  27.   How  to  train  pupils  to  read  problems. 

It  has  frequently  been  said  that  pupils  would  have  much  less 
difficulty  in  their  study  of  arithmetic  if  they  knew  how  to  read  prob- 
lems. Investigations  have  shown  that  children  are  very  careless  in 
the  reading  of  ordinary  prose.  Usually  they  are  not  able  to  read 
critically  and  carefully  when  the  material  is  very  simple.  However, 
arithmetic  problems  represent  a  special  type  of  reading  matter  and  a 
peculiar  type  of  comprehension  is  required. 

Corrective.  The  teacher  of  arithmetic  must  assume  some  of  the 
responsibility  for  training  pupils  to  read  the  problems.  It  is  fre- 
quently said  that  children  do  not  know  the  meanings  of  such  technical 
words  as  "factor,"  "product,"  "value,"  "per  pound,"  "are  obtained," 
etc.  These  words  define  relationships  which  exist  between  the  quan- 
tities and  are  cues  for  formulating  the  plan  of  solution.  The  teacher 
must  take  into  consideration  the  fact  that  the  vocabulary  is  different 
from  the  one  used  in  ordinary  conversation.  The  child  may  have 
heard  of  such  words  as  "average,"  "cost,"  and  "profit"  but  when 
asked  to  find  the  average,  to  find  the  cost,  etc.,  he  is  confronted  with 
a  problem  that  can  not  be  solved  from  his  general  knowledge.  To 
find  the  average  a  certain  definite  procedure  is  required.  The  vocab- 
ulary in  arithmetic  should  receive  the  same  treatment  as  a  vocabu- 
lary in  a  foreign  language,  i.  e.,  it  should  be  taught. 

One  teacher  stated  that  she  tried  to  get  her  pupils  to  read  the 
problems  orally  with  as  much  expression  as  they  would  use  in  read- 
ing a  story.  This  device  may  prove  helpful  in  many  cases  but,  in 
general,  problems  will  be  read  silently  rather  than  orally.  One  teacher 
frequently  reminded  her  pupils  that  a  "problem  always  tells  you 
directly  or  indirectly  what  you  are  to  do  if  you  understand  the  state- 
ment of  it." 

Difficulty  28.   How  to  teach  pupils  to  give  a  good  oral  explanation 
of  problems. 

Although  the  explanation  of  problems  has  occupied  a  very  prom- 
inent place  in  arithmetic  teaching,  it  is  not  inappropriate  to  ask  the 
question,  "Is  it  necessary  to  have  problems  explained  orally?"  The 
explanation  of  a  problem  is  justified  only  if  it  assists  the  pupil  giving 
the  explanation  or  his  classmates  in  learning  how  to  solve  the  prob- 
lem.  Whenever  the  explanation  becomes  merely  an  end  in  itself  and 

[33] 


is  mechanical  it  serves  no  useful  purpose.  The  writer  has  observed 
many  teachers  who  required  an  explanation  of  the  process  of  work- 
ing an  example  in  multiplication  or  division.  In  one  instance  all  the 
pupils  went  to  the  board  and  an  exercise  in  multiplication  was  given. 
All  performed  the  necessary  calculation  and  then  one  child  was  called 
upon  "to  explain  the  problem."  The  "problem"  (exercise)  was  to 
multiply  4569  by  43.  The  child  proceeded — "3  times  9  are  27,  put 
down  the  7  and  carry  the  2,  3  times  6  are  18  and  2  are  20,  put  down 
the  0  and  carry  the  2,  3  times  5  are  15  and  2  are  17,  put  down  the 
7  and  carry  the  1,"  etc.  It  is  doubtful  if  such  a  procedure  is  beneficial 
either  to  the  one  who  gives  the  explanation  or  to  those  who  listen. 

Corrective.  Instead  of  a  formal  explanation  the  teacher  and 
other  members  of  the  class  may  ask  the  pupil  why  he  did  this  or 
that.  In  the  case  of  a  problem  which  has  not  been  solved  by  other 
members  of  the  class,  the  pupil  should  feel  that  he  has  the  responsi- 
bility of  making  his  classmates  understand  how  he  solved  the  prob- 
lem and  the  reasons  for  his  procedure.  Occasionally  formal  explana- 
tions may  be  justified  but  not  as  a  usual  rule.  When  a  formal 
explanation  is  given  the  pupils  should  be  required  to  present  it 
effectively  and  to  use  good  English. 


[34] 


UNIVERSITY  OF  ILLINOIS-URBANA 


3  0112  065081942 


